created-9-24-04 - Limits Limit 1 lim n sin n 1 1 = lim x...

LimitsLimit # 1.limn→∞nsin1n= limx→∞xsin1x= limx→∞sin1x1x= limθ→0sinθθ= 1by Red #2, p 190.Definition 1.Asequenceis a function whose domain is the set of all positiveintegers.Definition 2.Let{an}be a sequence of real numbers and letLbe a real number.Then(a)limn→∞an=Liffthe following condition holds:For each open intervalUcentered atL,anis inUfrom some indexNonward.(b)limn→∞an=∞iff the following holds:For each intervalU= (M,∞],anis inUfrom some index onward.(c)lim

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